### Net present value (NPV) of next N years

The dividend discount model evaluates the value of all future dividends discounted to the present. This net present value can be compared to the stock price to evaluate whether the stock is overvalued or undervalued.

A nagging question emerges: since the DDM accounts for all the future dividends, is it really realistic? After all, the dividends after the next 100 years are not that useful in this lifetime. The stock could be sold anytime, of course, but even then one has to question the value of predicting too much into the future. A more conservative estimate would value the dividends in the intermediate future, say the next 30 to 50 years. Figure 1: Calculation of the present value of future dividends after N years.

Figure 1 shows the process of calculating the NPV of the dividends for the next N years. We already know the value of all the future dividends This same formula can be used to calculate the value of dividends N years into the future. Noting that that the future dividends have grown to DN = D0(1+g)N, the net future value (NFV) N years into the future is To reiterate: Equation (2) is the net future value of dividends still to come after N years have passed. Since we are not yet in the future, we need discount this future value to the present to obtain the present value of all the future dividends after N years
Equation (3) shows the net present value of all dividends after the next N years. The total NPV is the sum of next N years of dividends and dividends after the next N years: Equation (4) can be solved for the value of the next N years of dividends: Equation (5) simply states that the value of next N years of dividends is the difference between the total NPV given by Equation (1) and the NPV of dividends the after N given by Equation (3). By combining Equations (1), (3) and (5) we have: Table below shows a numerical example (Example#1) that is in line with the current value of SP500. The dividend is \$2 and it is expected to grow 5% per year. The long term discount rate is r = 7%. The NPV is \$107 and the fair yield is 2/107=1.9%. But as Table 1 shows, it takes a long time before this value is realized. The value of the next 40 years is barely half of the total value! The problem with the stock market is that the discount rate is close to the growth rate which leads to very long investment horizon. In comparison, we can compare the investment in a telecommunication company (Example#2) that has the dividend yield of Y=4.67% and the dividend growth rate of g=2%. The total discount rate is r = 8% beating the market average and the value is realized in a much shorter time span as shown in Table 2. More than half of the total value is realized in less than 20 years and the 80% of the value is realized in 30 years.

#### Value of next N years

Summary: Relying on growth rather than yield increases the investment horizon. The DDM shows that the current investment horizon for the stock market (SP500) is comparable to human lifetime. As such, index investing may not be attractive as a retirement savings instrument; For retirement, the timeline should measured in decades, not centuries.